For the pair of functions [tex]f(x)=\sqrt{\frac{x+4}{x+6}}[/tex] and [tex]g(x)=\sqrt{x-8}[/tex] find the following.
a. Find the functions [tex]f + g , f - g , fg[/tex], and [tex]\frac{ f }{ g }[/tex].
b. Determine the domain of the functions [tex]f + g , f - g , fg[/tex], and [tex]\frac{ f }{ g }[/tex].
a. [tex](f+g)(x)=\sqrt{\frac{x+4}{x+6}}+\sqrt{x-8}[/tex]
(Do not simplify. Type an exact answer, using radicals as needed. Do not rationalize the denominator.)
[tex](f-g)(x)=\sqrt{\frac{x+4}{x+6}}-\sqrt{x-8}[/tex]
(Do not simplify. Type an exact answer, using radicals as needed. Do not rationalize the denominator.)
[tex](f g)(x)=\sqrt{\frac{(x+4)(x-8)}{x+6}}[/tex]
(Simplify your answer. Type an exact answer, using radicals as needed. Do not rationalize the denominator.)
[tex]\left(\frac{f}{g}\right)(x)=[/tex]
[tex]\square[/tex]
(Simplify your answer. Type an exact answer, using radicals as needed. Do not rationalize the denominator.)
Answer (2)
Find the expression for g f : ( g f ) ( x ) = ( x + 6 ) ( x − 8 ) x + 4 .
Determine the domain of f ( x ) : ( − ∞ , − 6 ) ∪ [ − 4 , ∞ ) .
Determine the domain of g ( x ) : [ 8 , ∞ ) .
The domain of f + g , f − g , and f g is [ 8 , ∞ ) , and the domain of g f is ( − 6 , − 4 ] ∪ ( 8 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 6 x + 4 and g ( x ) = x − 8 , and we need to find the expressions for f + g , f − g , f g , and g f , as well as their respective domains.
Finding f/g First, let's find the expression for g f : g ( x ) f ( x ) = x − 8 x + 6 x + 4 = ( x + 6 ) ( x − 8 ) x + 4 So, ( g f ) ( x ) = ( x + 6 ) ( x − 8 ) x + 4
Domain of f(x) Now, let's determine the domain of f ( x ) . We need x + 6 x + 4 ≥ 0 . This inequality holds when both the numerator and the denominator are positive or both are negative.
Case 1: x + 4 ≥ 0 and 0"> x + 6 > 0 . This gives x ≥ − 4 and -6"> x > − 6 . The intersection is x ≥ − 4 .
Case 2: x + 4 ≤ 0 and x + 6 < 0 . This gives x ≤ − 4 and x < − 6 . The intersection is x < − 6 .
So, the domain of f ( x ) is ( − ∞ , − 6 ) ∪ [ − 4 , ∞ ) .
Domain of g(x) Next, let's determine the domain of g ( x ) . We need x − 8 ≥ 0 , which means x ≥ 8 . So, the domain of g ( x ) is [ 8 , ∞ ) .
Domain of f+g, f-g, and fg Now, we find the domain of f ( x ) + g ( x ) , f ( x ) − g ( x ) , and f ( x ) g ( x ) . This is the intersection of the domains of f ( x ) and g ( x ) .
So, we have (( − ∞ , − 6 ) ∪ [ − 4 , ∞ )) ∩ [ 8 , ∞ ) = [ 8 , ∞ ) .
Domain of f/g Finally, let's determine the domain of g ( x ) f ( x ) . We need ( x + 6 ) ( x − 8 ) x + 4 ≥ 0 and g ( x ) = 0 , which means 0"> x − 8 > 0 , so 8"> x > 8 .
We analyze the sign of ( x + 6 ) ( x − 8 ) x + 4 . The critical points are − 6 , − 4 , and 8 .
If x < − 6 , then x + 4 < 0 , x + 6 < 0 , and x − 8 < 0 . So, ( x + 6 ) ( x − 8 ) x + 4 < 0 .
If − 6 < x < − 4 , then x + 4 < 0 , 0"> x + 6 > 0 , and x − 8 < 0 . So, 0"> ( x + 6 ) ( x − 8 ) x + 4 > 0 .
If − 4 < x < 8 , then 0"> x + 4 > 0 , 0"> x + 6 > 0 , and x − 8 < 0 . So, ( x + 6 ) ( x − 8 ) x + 4 < 0 .
If 8"> x > 8 , then 0"> x + 4 > 0 , 0"> x + 6 > 0 , and 0"> x − 8 > 0 . So, 0"> ( x + 6 ) ( x − 8 ) x + 4 > 0 .
Since we need ( x + 6 ) ( x − 8 ) x + 4 ≥ 0 , the solution is ( − 6 , − 4 ] ∪ ( 8 , ∞ ) .
Final Domains Therefore, the domain of f + g , f − g , and f g is [ 8 , ∞ ) , and the domain of g f is ( − 6 , − 4 ] ∪ ( 8 , ∞ ) .
Examples
Understanding the domains of combined functions is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the velocity of an object and g ( x ) represents the time it takes to travel a certain distance, then the domain of f ( x ) / g ( x ) would tell us the valid range of times for which the velocity calculation makes sense. Similarly, in economics, if f ( x ) represents the revenue and g ( x ) represents the cost, then the domain of f ( x ) − g ( x ) would indicate the range of production levels for which the profit is non-negative.
The expressions for the combination of functions f + g , f − g , f g , and g f have been determined, along with their respective domains. The domains for f + g , f − g , and f g is [ 8 , ∞ ) , while for g f , it is ( − 6 , − 4 ] ∪ ( 8 , ∞ ) .
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